Riemann Zeta Function at Even Integers/Lemma

From ProofWiki
Jump to navigation Jump to search

Lemma

Let $x \in \R$ be such that $\size x < 1$.

Then:

$\ds \pi x \cot {\pi x} = 1 - 2 \sum_{n \mathop = 1}^\infty \map \zeta {2 n} x^{2 n}$

where $\zeta$ denotes the Riemann zeta function.


Proof 1

\(\ds \frac {\sin \pi x} {\pi x}\) \(=\) \(\ds \prod_{k \mathop = 1}^\infty \paren {1 - \frac {x^2} {k^2} }\) Euler Formula for Sine Function
\(\ds \leadsto \ \ \) \(\ds \map \ln {\frac {\sin \pi x} {\pi x} }\) \(=\) \(\ds \ln \prod_{k \mathop = 1}^\infty \paren {1 - \frac {x^2} {k^2} }\)
\(\ds \leadsto \ \ \) \(\ds \map \ln {\sin {\pi x} }\) \(=\) \(\ds \map \ln {\pi x} + \sum_{k \mathop = 1}^\infty \map \ln {1 - \frac {x^2} {k^2} }\) Laws of Logarithms
\(\ds \leadsto \ \ \) \(\ds \pi \frac {\cos {\pi x} } {\sin {\pi x} }\) \(=\) \(\ds \frac 1 x + \sum_{k \mathop = 1}^\infty \frac 1 {\paren {1 - \frac {x^2} {k^2} } } \paren {-\frac {2 x} {k^2} }\) differentiating with respect to $x$
\(\ds \leadsto \ \ \) \(\ds \pi x \cot {\pi x}\) \(=\) \(\ds 1 + \sum_{k \mathop = 1}^\infty \frac 1 {\paren {1 - \frac {x^2} {k^2} } } \paren {-\frac {2 x^2} {k^2} }\)
\(\ds \) \(=\) \(\ds 1 + \sum_{k \mathop = 1}^\infty \sum_{n \mathop = 0}^\infty \paren {\frac {x^2} {k^2} }^n \paren {-\frac {2 x^2} {k^2} }\) Sum of Infinite Geometric Sequence
\(\ds \) \(=\) \(\ds 1 - 2 \sum_{k \mathop = 1}^\infty \sum_{n \mathop = 1}^\infty \paren {\frac {x^2} {k^2} }^n\)
\(\ds \) \(=\) \(\ds 1 - 2 \sum_{n \mathop = 1}^\infty \paren {\sum_{k \mathop = 1}^\infty \frac 1 {k^{2 n} } } x^{2 n}\) interchanging order of summation is valid by Tonelli's Theorem
\(\ds \) \(=\) \(\ds 1 - 2 \sum_{n \mathop = 1}^\infty \map \zeta {2 n} x^{2 n}\)

$\blacksquare$


Proof 2

From Laurent Series Expansion for Cotangent Function:

$\ds \pi \cot \pi z = \frac 1 z - 2 \sum_{n \mathop = 1}^\infty \map \zeta {2 n} z^{2 n - 1}$

where:

$z \in \C$ such that $\cmod z < 1$
$\zeta$ is the Riemann Zeta function.


Letting $x \in \R$ replace $z$, and multiplying through by $x$:

\(\ds \pi x \cot \pi x\) \(=\) \(\ds x \paren {\frac 1 x - 2 \sum_{n \mathop = 1}^\infty \map \zeta {2 n} x^{2 n - 1} }\)
\(\ds \) \(=\) \(\ds 1 - 2 x \sum_{n \mathop = 1}^\infty \map \zeta {2 n} x^{2 n - 1}\)
\(\ds \) \(=\) \(\ds 1 - 2 \sum_{n \mathop = 1}^\infty \map \zeta {2 n} x^{2 n}\)

$\blacksquare$